Topological conditions for the unique solvability of linear time-invariant and time-varying networks

The problem of knowing whether the non-unique solvability depends on the particular values of the components or on their topological interconnections is studied for linear networks with arbitrary, time-invariant as well as time-varying n-ports. Within every network, the topological notions of its sockets and of their independence are introduced. Networks with independent sockets are shown-at least when there are no relations among the non-zero coefficients, nor repetitions of the same coefficient are allowed, i.e. under suitable generality assumptions-to be uniquely solvable. Networks with dependent sockets are shown to be never uniquely solvable. Polynomially bounded algorithms, requiring only integer arithmetic, to test independence are available. When independence fails, a topological configuration of components which shows fewer topologically independent variables than equations, is proved to exist.     

Sufficient conditions for the unique solvability of linear networks containing memoryless 2-ports

Combinatorial necessary and sufficient conditions for the unique solvability of linear networks containing n-ports are well known for the ‘general’ case. They are only necessary if relations among n-port parameters are also taken into consideration. In the present paper combinatorial sufficient conditions are presented for linear networks containing RLC elements and memoryless 2-ports. The somewhat surprising result is proved that whether a 2-port can cause certain types of singularities can be predicted before the interconnection. A concept, similar to the normal tree (which intersects ideal transformers by one, gyrators by two or no edges) is introduced for arbitrary 2-ports. Its existence implies unique solvability. Relations to previous results and algorithmical aspects are also discussed.     

The generalized tree for state variables in linear active networks

A tree for the association of state variables in linear active networks is presented. This tree is selected in such a manner that the voltages across the capacitors contained in it and the currents through the inductors contained in its cotree may be chosen as state variables of a linear network. An algorithm for finding such a tree is given, and the association of state variables with this tree are demonstrated by examples.     

Determination of initial conditions in active RLC networks

A new method is described of determining the initial values of state variables in active RLC networks. This method utilizes neither the conservation principle of charges and fluxes nor the theory of distributions. An RLC active network is assumed to be formed at t = 0 by joining or disconnecting the elements that are contained in it, this operation being performed by the use of a set of switches. The new method is based on the consideration of switches as independent sources, and the state equations are extended to cover this situation, and their solution at t = 0⁺ gives the initial conditions that have been sought. Further, the new method is applicable to any state-space formulation techniques used.     

Some considerations on state equations of linear active networks

It is shown that if there equivalently exist the virtual resistive elements in parallel with the inductive elements of the over-normal tree of a given linear active network or in series with the capacitive elements of the corresponding co-tree, an increase in the number of state variables arises. It is also shown that when a virtual resistor equivalently appears in parallel with the distinct resistor in a tree or in series with the distinct resistor in the co-tree, a decrease in the number of state variables may arise. This is, however, a rare case in the usual types of network. Two algebraic methods for obtaining the state equation of linear active networks are presented. One is useful for the networks in which the decrease in the number of state variables does not arise. From the other, the output equation for the required variables is obtained at the same time as the state equation. Further, the initial values are simply determined without iteration in many cases.     

Unique solvability of RLC nullor networks

Sufficient conditions for the unique solvability of linear active networks modelled as RLC nullor networks are given. an interpretation of these sufficient conditions in terms of subnetworks of the network being studied is also given. A simple procedure for calculating the matrices involved in these sufficient conditions is described and illustrated by example.     

Topological transformations of electrical networks

This paper deals with effects of modifications of network structure that may be studied without reference to the type of devices present in the network. We introduce and make systematic use of the notion of a generalized minor of a vector space. This operation generalizes the usual short and open circuit operations for a graph. Using the generalized minor operation we show how to make the equations of a given network appear to be the ‘bordered version’ of the equations of some other specified network. We also consider the decomposition of a network into several ‘multiports’ and a ‘port connection diagram’, and study the properties of a minimal decomposition (with port connection diagram having a minimum number of edges). In each case we present efficient algorithms wherever appropriate. Although the paper makes use of ideas from elementary matroid theory it is entirely self-contained and requires no more than the knowledge of elementary linear algebra from the reader.     

On the existence of solutions to linear active networks: A state-space approach

In recent literature, several alternative conditions for the existence of solutions to active networks are given. In this paper, yet another condition is given which is both necessary and sufficient for the existence of a unique solution to the network. Based on this new condition a precise upper bound for the order of complexity of an active network is established which differs from published results. A Fortran coded program is also available.     

Topological properties of high-voltage electrical transmission networks

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.     

On the solvability and computation of D.C. transistor networks

A study of non-linear d.c. networks containing transistors, diodes, linear resistors, independent voltage and current sources and linear controlled voltage and current sources, described by Tf(x) + Gx = b is presented. A few theorems concerning both the solvability and numerical computation of d.c. transistor networks are proved. The conditions sufficient for the existence of at least one solution and of a unique solution are defined. A method for numerical computation of the networks without the need to determine inverse matrices is presented and the convergence of the iterative technique is analysed. Suggestions are formulated regarding applicability of the method for computation of an approximate solution (close to the exact solution) which may subsequently be quickly corrected using the Newton-Raphson method.     

Using the superposition method to formulate the state variablematrix for linear networks

This paper proposes a systematic method for writing the state variable matrix for nondegenerate linear electrical networks by using the superposition principle and also extends the procedure to cover degenerate networks. To use the proposed method, all inductances and capacitances in a network are replaced by ideal independent sources; the state variable equations are then written, often by inspection with nondegenerate networks, utilizing the superposition principle in conjunction with the resulting simplified electrical network     

Realizability conditions on short-circuit conductance matrix by means of grounded active resistive networks embedding grounded VCVSs

The proof is given of a new theorem concerning the minimum number of ideal grounded voltage-controlled voltage sources (VCVSs) necessary and sufficient to realize an arbitrary real square matrix as the short-circuit conductance matrix of a grounded transformerless active resistive multiport network embedding grounded VCVSs.     

Stable high q active networks

A new approach is suggested to approximate the amplitude frequency behaviour of a given high Q poles network function, by an explicit function having poles with a predetermined desired low Q, and with explicit knowledge and control of the error. The method is practical for reducing Q's by the ratio of up to 6. Any critical pole-pair is replaced by an approximating function possessing a complex conjugate n-th order pole-pair and a (n?1)-th order zero-pair. A general second order low Q building block is derived as an example. Measured results proving the practical usefulness of the method, are reported.     

The equidistant linear phase polynomial for distributed and digital networks

A closed form expression for the recurrence formula for the generation of polynomials whose phase interpolates to linear characteristics at equidistant frequencies is presented. This polynomial may be used directly in the design of both distributed and digital networks and, due to the recurrence formula, readily generated. Since the complete proof of this formula is extensive and not required for normal use of this polynomial, it has been outlined in an Appendix.     

直线电机的线性自抗扰控制

光刻机的工件台是高动态精密伺服运动平台,它要求在系统高速运动的同时,采用长行程直线电机宏动跟随音圈电机高精密微动的驱动方式,实现系统纳米级的精确定位及跟踪。为减小微动电机的运动范围和加速度,必须提高直线电机的跟踪精度。针对该系统的直线电机模型设计了一种线性自抗扰控制方法,该方法的控制器首先通过扩张状态观测器观测系统的动态变化,补偿系统中的各种扰动,再运用前馈对系统的跟踪误差进行补偿,减小系统的动态跟踪误差。在此复合控制方式下,控制器实现了自抗扰控制,前馈控制器很好的补偿了误差,从而提高了系统的抗干扰和跟踪性能。实验表明,该方法与传统的控制方法相比,改善了系统的动态性能和抗干扰能力。     

Necessary and sufficient conditions for absolute stability of linear n-ports

A set of necessary and sufficient conditions is stated and proved for the absolute stability (under any passive terminations), in the ‘bibo’ sense, of a linear n-port characterized by its open-circuit impedance matrix. A more explicit set of such conditions is derived for the special case n = 3. In the process of deriving this set of conditions, some results and theorems concerning finding the zero sets of two and three parameter functions, are stated and proved. Also derived is a set of necessary and sufficient conditions for the absolute stability of two-ports characterized by the scattering parameters. Such sets have already been derived previously, but the present set is considerably simpler.